Problem: Given triangle ABC with side lengths a, b, and c. Let D, E, and F be the points of tangency of the incircle, as shown. Construct triangle DEF.

(a) Prove that triangle DEF is acute, that is, that the triangle determined by the points of tangency of the incircle is always acute.

(b) Find the area of triangle DEF in terms of a, b, and c.

(c) Prove that AD, BF, and CE are concurrent. Explore this point of concurrency as the shape of triangle ABC is varied. (NOTE: This point of concurrency is NOT the incenter except in particular cases. When would this point of concurrency be the incenter?)



Click here for a GSP sketch of the configuration.